Differential Transformations of Parabolic Second-Order Operators in the Plane

Darboux's classical results about transformations of second-order hyperbolic equations by means of differential substitutions are extended to the case of parabolic equations of the form $Lu=(D^2_x+a(x,y)D_x+b(x,y)D_y+c(x,y))u=0$. We prove a general theorem that provides a way to determine admissible differential substitutions for such parabolic equations. It turns out that higher order transforming operators can always be represented as a composition of first-order operators that define a series of consecutive transformations. The existence of inverse transformations imposes some differential constrains on the coefficients of the initial operator. We show that these constraints may imply famous integrable equations, in particular, the Boussinesq equation.